Sampled STFT phase retrieval: foundational discretization barriers and relations to the completeness problem of discrete translates

23.05.2022 15:00 - 16:30

Lukas Liehr (University of Vienna)

The STFT phase retrieval problem concerns the reconstruction of a square-integrable function f from the absolute value of its short-time Fourier transform (the so-called spectrogram) with respect to a window function g, sampled at L, i.e. from the set |Vgf(z)| = { |Vgf(z)| : z ∈ L}. For which window functions g and for which sampling sets L is every f determined (up to a global phase) by |Vgf(L)|? In view of practical applications, this question is of major significance in the situation where L is a lattice.
The first part of the talk concentrates on the univariate case d=1. We prove that there exists no window function g and no lattice L such that every f is determined up to a global phase from its spectrogram samples at L. This result highlights a stark contrast to classical results in time-frequency analysis where phase information is present. Following this line of research, we consider the multivariate case d>1 and show that similar non-uniqueness statements hold true for a large class of lattices, prime examples include symplectic lattices and separable lattices. We conclude the talk with a discussion of a restricted form of the STFT phase retrieval problem, namely the recovery of compactly supported signals. In this case, we relate the problem to the completeness problem of discrete translates and obtain uniqueness results from spectrogram samples given on separable lattices.

Joint work with Philipp Grohs.

M. Faulhuber and K. Gröchenig
SR10 (2nd floor)